Editing Divergence (section)

== Physical interpretation of divergence ==
{{See also|sources and sinks}}
In physical terms, the divergence of a vector field is the extent to which the vector field [[flux]] behaves like a [[sources and sinks|source or a sink]] at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it.   A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field.  A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field.  The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point.  A point at which there is zero flux through an enclosing surface has zero divergence.

The divergence of a vector field is often illustrated using the simple example of the [[velocity field]] of a fluid, a liquid or gas.  A moving gas has a [[velocity]], a speed and direction at each point, which can be represented by a [[vector (mathematics and physics)|vector]], so the velocity of the gas forms a [[vector field]]. If a gas is heated, it will expand.  This will cause a net motion of gas particles outward in all directions.  Any closed surface in the gas will enclose gas which is expanding, so there will be an outward flux of gas through the surface.  So the velocity field will have positive divergence everywhere.  Similarly, if the gas is cooled, it will contract.  There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface.  Therefore, the velocity field has negative divergence everywhere.  In contrast, in a gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero.  The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the ''net'' flux is zero. Thus the gas velocity has zero divergence everywhere.  A field which has zero divergence everywhere is called [[solenoidal vector field|solenoidal]].

If the gas is heated only at one point or small region, or a small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point.  Any closed surface enclosing the heated point will have a flux of gas particles passing out of it, so there is positive divergence at that point.  However any closed surface ''not'' enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero.  Therefore, the divergence at any other point is zero.